Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 269 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 269 }$: $ x^{2} + 268 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 199 a + 208 + \left(159 a + 69\right)\cdot 269 + \left(80 a + 125\right)\cdot 269^{2} + \left(51 a + 121\right)\cdot 269^{3} + \left(9 a + 117\right)\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 238 a + 162 + \left(142 a + 241\right)\cdot 269 + \left(231 a + 231\right)\cdot 269^{2} + \left(120 a + 180\right)\cdot 269^{3} + \left(128 a + 131\right)\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 a + 138 + \left(109 a + 30\right)\cdot 269 + \left(188 a + 46\right)\cdot 269^{2} + \left(217 a + 92\right)\cdot 269^{3} + \left(259 a + 75\right)\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 130 a + 162 + \left(86 a + 144\right)\cdot 269 + \left(3 a + 158\right)\cdot 269^{2} + \left(32 a + 165\right)\cdot 269^{3} + \left(222 a + 227\right)\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 253 + 72\cdot 269 + 118\cdot 269^{2} + 251\cdot 269^{3} + 235\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 139 a + 23 + \left(182 a + 101\right)\cdot 269 + \left(265 a + 75\right)\cdot 269^{2} + \left(236 a + 194\right)\cdot 269^{3} + \left(46 a + 148\right)\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 31 a + 131 + \left(126 a + 146\right)\cdot 269 + \left(37 a + 51\right)\cdot 269^{2} + \left(148 a + 70\right)\cdot 269^{3} + \left(140 a + 139\right)\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
